Deriving Homoskedastic 2SLS Variance

Why 2SLS Needs Its Own Variance Calculation

Two-stage least squares (2SLS) uses the variation in regressors projected onto the instrument space. Its coefficient estimate may be computed through a fitted-regressor representation, but that does not mean the usual second-stage OLS standard error is automatically valid. To obtain the conventional homoskedastic 2SLS variance, we use the 2SLS estimator itself and the assumed covariance structure of the structural errors.

This tutorial derives the finite-sample variance formula under explicit conditional assumptions, then shows how its unknown error variance is conventionally estimated. The same matrix expression is also the familiar large-sample homoskedastic variance estimator under standard IV moment conditions and a constant conditional second moment. Those statements concern related formulas, but their assumptions should not be conflated.

The matrix \(P_Z\) is symmetric and idempotent: \(P_Z'=P_Z\) and \(P_ZP_Z=P_Z\). The projected regressor matrix is \(\hat X=P_ZX\). Thus \(X'P_ZX=\hat X'\hat X\), and the 2SLS estimator can be written as \((\hat X'\hat X)^{-1}\hat X'y\). This representation is useful for deriving the variance, but \(\hat X\) is constructed from instruments and is not a license to treat the second stage as ordinary OLS with fixed regressors in every respect.

Assumptions for the Homoskedastic Formula

For the exact conditional variance derivation below, regard the design matrices \(X\) and \(Z\) as fixed, or condition on them. Require that \(X'P_ZX\) is nonsingular and that the errors have conditional mean zero and conditional covariance matrix \(\sigma^2 I_n\). The latter says that, given the design, each error has variance \(\sigma^2\) and distinct errors have zero conditional covariance. Conditional independence with a common finite variance is one sufficient way to obtain this covariance structure; independence is stronger than the derivation requires.

A related asymptotic formulation is worth distinguishing. With independent observations and the IV moment condition \(E[Z_i u_i]=0\), a constant conditional second moment \(E[u_i^2\mid Z_i]=\sigma^2\) yields the homoskedastic simplification of the large-sample IV sandwich variance. Here “second moment” is deliberate: it is not generally the same as conditional variance. The equality \(E[u_i^2\mid Z_i]=\operatorname{Var}(u_i\mid Z_i)\) holds only if \(E[u_i\mid Z_i]=0\). The finite-sample derivation in the next section instead explicitly assumes conditional mean zero and a conditional covariance matrix of \(\sigma^2 I_n\).

Deriving the Conditional Variance

Substitute the structural equation \(y=X\beta+u\) into the 2SLS estimator. The resulting estimation error is a linear transformation of \(u\). Applying the conditional covariance rule to that transformation gives the variance formula.

Because \(E[u\mid X,Z]=0\), the conditional expectation of \(\hat\beta_{2SLS}\) is \(\beta\). Its conditional variance is therefore the conditional covariance of the linear error term. Write \(A=(X'P_ZX)^{-1}X'P_Z\). Then \(\hat\beta_{2SLS}-\beta=Au\), so \(\operatorname{Var}(\hat\beta_{2SLS}\mid X,Z)=A\operatorname{Var}(u\mid X,Z)A'\). Use \(\operatorname{Var}(u\mid X,Z)=\sigma^2I_n\), the symmetry of \(P_Z\), and its idempotence:

The last equality uses the nonsingularity of \(X'P_ZX\). This derivation is the reason the projected-regressor matrix appears in the conventional variance formula: \(P_ZP_Z=P_Z\) reduces the middle product to \(X'P_ZX\). The result is a \(K\times K\) matrix. Its diagonal entries are conditional variances of the coefficient estimates; off-diagonal entries are their conditional covariances.

From the Formula to a Standard Error

The error variance \(\sigma^2\) is unknown. A conventional estimate uses the structural residuals \(\hat u=y-X\hat\beta_{2SLS}\) and sets \(\hat\sigma^2=\hat u'\hat u/(n-K)\). The resulting homoskedastic covariance estimator is \(\widehat{\operatorname{Var}}(\hat\beta_{2SLS})=\hat\sigma^2(X'P_ZX)^{-1}\). This residual-based estimate of \(\sigma^2\) is commonly used and is consistent under standard conditions, but with 2SLS it is not generally an exactly unbiased finite-sample estimate. The degrees of freedom \(n-K\) account for the number of structural coefficients estimated.

How Projected Variation Affects Precision

In the scalar case, the variance is \(\sigma^2/(x'P_Zx)\). Holding the error variance fixed, a larger \(x'P_Zx\) means a smaller conditional variance. Because \(x'P_Zx=(P_Zx)'(P_Zx)\), this denominator measures the squared length of the component of the regressor that lies in the instrument space. It is not simply the total variation in \(x\); variation in \(x\) outside that space does not contribute to this denominator.

Common Mistakes and Scope

• Using the second-stage OLS standard error without justification. The conventional 2SLS variance follows from the 2SLS estimator and the assumed error covariance structure. Treating fitted regressors as if they were ordinary observed regressors does not establish that the usual second-stage OLS standard error is valid.
• Calling a conditional second moment a conditional variance. \(E[u_i^2\mid Z_i]=\sigma^2\) is a constant conditional second-moment assumption. It equals \(\operatorname{Var}(u_i\mid Z_i)\) only when \(E[u_i\mid Z_i]=0\). Instrument validity in the form \(E[Z_i u_i]=0\) does not by itself supply that conditional mean-zero condition.
• Assuming the residual-based \(\hat\sigma^2\) is exactly unbiased. The conventional estimator \(\hat u'\hat u/(n-K)\) is a standard plug-in estimate, but 2SLS residuals do not generally make this exact finite-sample unbiasedness claim true.
• Reading \(x'P_Zx\) as total regressor variation. It is the squared length of the projected regressor. The formula’s precision depends on instrument-projected variation, not on every component of \(x\).

The homoskedastic expression is appropriate only when its covariance assumptions are defensible for the analysis. If error variability differs across observations, the simplification to \(\hat\sigma^2(X'P_ZX)^{-1}\) need not provide the appropriate variance estimate. The central lesson here is to connect the standard error to the assumptions that produced it, rather than selecting it solely because a software output labels it “2SLS.”